For a given –th root of unity
, we give explicit formulas
of a family of –variable
Laurent polynomials
with coefficients in that
encode the –symbols
associated with nilpotent representations of
. For a given
abelian group ,
we use them to produce a state sum invariant
of a quadruplet
(compact –manifold
, link
inside
, homology class
, homology
class ) with
values in a ring
related to .
The formulas are established by a “skein” calculus as an application
of the theory of modified dimensions introduced by the authors and
Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented
–manifold
, the invariants are
related to defined
by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of
. They refine
them as where
correspond
to with the
isomorphism .
Keywords
$6j$–symbols, state sum, skein calculus, quantum groups,
$3$–manifolds
